The full calculation from this geometric idea was explained in [1]}. It is easy however, from this perspective, to re-interpret the formulae obtained in the quantum context, in order to explain the classical result. In the quantum context, equation (REF ) is the free energy density for a system of free particles where each quasi-momentum \(p\)  is independently distributed in accordance with the Boltzmann weight determined by the “energy" \(\epsilon (p)\) . But the Yang-Yang equation (REF ) says that the interaction “dresses" this energy: it modifies the probability for the level to be occupied by taking into account the free energy within the additional space introduced by mode \(p\) .
